We first present one proposition.
Since \(d_i=\sum ^{}_{k}a_{i,k}\)  and \(b_j=\sum ^{}_{k}a_{k,j}\)  are sums of \(m\)  and \(n\)  independent random variables,
by the central limit theorem for the bounded case in Loève ([1]}, page 289), we know that \({v_{i,i}}^{-1/2}(d_i-\mathbb {E}(d_i))\)  and \({v_{m+j,m+j}}^{-1/2}(b_j-\mathbb {E}(b_j))\)  are asymptotically standard normal if \(v_{i,i}\)  and \(v_{m+j,m+j}\)  diverge, respectively.
Note that
\(\frac{m e^{2\parallel {{\theta }^{*}}\parallel _{\infty }}}{(1+e^{2\parallel {{\theta }}^*\parallel _{\infty }})^2}\le v_{i,i}\le \frac{m}{4},~~\frac{n e^{2\parallel {{\theta }^{*}}\parallel _{\infty }}}{(1+e^{2\parallel {{\theta }}^*\parallel _{\infty }})^2}\le v_{m+j,m+j}\le \frac{n}{4}.\) 
